Coulomb-Frustrated Phase Separation Phase Diagram in Systems with Short-Range Negative Compressibility

Using numerical techniques and asymptotic expansions we obtain the phase diagram of a paradigmatic model of Coulomb-frustrated phase separation in systems with negative short-range compressibility. The transition from the homogeneous phase to the inhomogeneous phase is generically first order in isotropic three-dimensional systems except for a critical point. Close to the critical point, inhomogeneities are predicted to form a bcc lattice with subsequent transitions to a triangular lattice of rods and a layered structure. Inclusion of a strong anisotropy allows for second- and first-order transition lines joined by a tricritical point.

Figure 1

Phase diagram in three-dimensional isotropic systems. The small dots indicate the Gaussian instability line $Q_g$. The thin (thick) lines represent first-order transitions in the strong (weak) coupling approximation. In the two limits they overlap with the corresponding numerically determined transition lines from homogeneous to dropletlike inhomogeneities (□), from droplets to rods (▵), and from rods to layers (⋄).

Figure 2

(a) Behavior of the charge density modulation for $\overline{\varphi }=0$ and different couplings (labeled by the value of $Q/Q_c$). Near the Gaussian instability the modulation is close to a SCDW (thin full line). As $Q$ decreases, unharmonicity is built in and the charge modulation tends to a square wave which has, as limiting case, Maxwell construction at $Q=0$. (b) Evolution of the periodicity of the charge density wave measured in units of $\xi$. Inset: Evolution of the harmonic amplitudes ${\varphi }_n/\sum _{n^{\prime }>0}{\varphi }_{n^{\prime }}$ as a function of $Q$. The dots indicate the amplitudes for a layered structure of $\varphi =\pm 1$ as ruled by Maxwell construction at $Q=0$. Amplitudes for even $n$ vanish by symmetry.

Figure 3

The phase diagram for anisotropic 3D systems. The thin line corresponding to the Gaussian instability determines the second-order transition line above the tricritical point (●). Below the tricritical point the transition is first order and is determined numerically (□) and in the weak coupling limit assuming sharp interfaces [2] (thick line). Below the tricritical point the Gaussian line becomes the limit of metastability of the uniform phase.